Lagrange's theorem gives a strong test for possible subgroup orders. Once the order of a finite group is known, the order of every subgroup must be one of its positive divisors. This lecture focuses on that restriction. We will not claim that every divisor gives a subgroup; instead, we learn how Lagrange's theorem rules out impossible subgroup orders. This is a basic but essential skill in finite group theory.

Lagrange's theorem gives a divisibility filter for subgroup orders. Choose the order of a finite group and a candidate subgroup order, then observe which positive divisors survive the test. A highlighted divisor means the order is allowed by Lagrange's theorem, not guaranteed to occur. A non-divisor is ruled out immediately.

The filter is one-way. Passing the divisibility test only says that Lagrange's theorem allows the order; failing the test proves that the subgroup order is impossible.

This corollary is often used negatively. For example, if a group has order $30$, then it cannot have a subgroup of order $8$, because $8$ does not divide $30$. The theorem does not say that a group of order $30$ must have subgroups of orders $2,3,5,6,10,$ or $15$. Those are only possible orders by divisibility.

This calculator checks a proposed subgroup order against a known finite group order. Enter $|G|$ and a candidate $d$ to test whether $d$ divides $|G|$. When the candidate passes, the calculator also gives the possible index $|G|/d$. When it fails, Lagrange's theorem rules out that subgroup order completely.